Dynamic Modeling and Analysis of an Omnidirectional Mobile Robot

2013 IEEE/RSJ International Conference on Intelligent Robots and Systems (IROS) November 3-7, 2013. Tokyo, Japan

Dynamic Modeling and Analysis of an Omnidirectional Mobile Robot

Chao Ren1 and Shugen Ma2

Abstract-- This paper presents the dynamic modeling and analysis of a three-wheeled omnidirectional mobile robot with MY wheels-II, whose dynamics is nonlinear and piecewisesmooth. Firstly, the detailed dynamic model of the robot is derived, which shows that the robot is actually a switched nonlinear system. Analysis of the robot dynamic properties based on the detailed dynamic model is presented in detail. Then to facilitate the controller design for the switched nonlinear system, based on the detailed dynamic model, an average dynamic model is proposed by simply averaging the wheel contact radius. The resulting average dynamic model is nonlinear and smooth, which may then be used as one solution for the model-based control design. Open-loop simulation results show the dynamic properties of the mobile robot. In addition, the effectiveness of the proposed average model in predicting characteristics of the detailed dynamic model is also illustrated through open-loop simulations.

I. INTRODUCTION

Omnidirectional mobile robots (OMRs) are becoming increasingly popular in many applications, especially those in the narrow spaces, since they can perform both translational and rotational motion independently and simultaneously. In other words, they can move in any direction with any orientation angle.

Various omnidirectional wheel mechanisms were proposed in the past few decades. These mechanisms can be divided into two groups: non-switch wheels and switch wheels, depending on whether the contact radius of the wheel to the robot gravity center switches. Majority of the wheels, such as Mecanum wheel [1], Alternate wheel [2] and Ball wheel [3], are of the first group. The five switch wheel mechanisms proposed until now are shown in Fig. 1. They are Longitudinal Orthogonal-wheel [4], MY wheel [5], MY wheel-II [6], Swedish wheel [7] and Omni-wheel [8]. The dynamics of the non-switch wheeled OMRs is nonlinear and smooth while the switch wheeled OMRs are nonlinear and piecewise-smooth dynamical systems.

In our previous study, we proposed two switch wheel mechanisms, namely MY wheel [5] and MY wheel-II [6], and developed the prototype platforms. The proposed MY wheel mechanisms have several advantages over traditional wheel mechanisms, such as high payload and insensitive to fragments on the ground. However, in practice, we found that it is extremely difficult to design the controller directly based on the dynamic model due to its hybrid nature. In addition, all of the previous researches on the dynamic modeling and control are for the non-switch wheeled and Swedish wheeled

C. Ren is with Department of Robotics, Ritsumeikan University, 5258577, Shiga, Japan gr0119vp@ed.ritsumei.ac.jp

S. Ma is with Department of Robotics, Ritsumeikan University, 525-8577, Shiga, Japan shugen@se.ritsumei.ac.jp

(a)

(b)

(c)

(d)

(e)

Fig. 1. Switch wheel mechanisms: (a) Longitudinal Orthogonal-wheel. (b) MY wheel. (c) MY wheel-II. (d) Swedish wheel. (e) Omni-wheel.

or Omni-wheeled OMRs, based on continuous dynamic models [9]?[13], to name a few. For the Swedish wheeled or Omni-wheeled OMRs, it is worth pointing out that the continuous dynamic models were directly employed in the previous studies by regarding it as a non-switch wheel, but no analysis was presented about the resulting modeling errors due to neglecting the switching effects.

For the switch wheeled OMRs, kinematic modeling and analysis were studied in [4], [6], [14], [15]. In [4], the kinematic model was derived for a mobile platform with three Longitudinal Orthogonal-wheel assemblies. For the same kind of omnidirectional mobile platform in [14], the average wheel contact radius was used instead of the real contact radius in the inverse kinematic model, to solve the problem of motor angular velocity fluctuations. In [6] and [15], the kinematic analysis was studied by using an optimal scale factor (OSF) instead of the average contact radius and the factors influencing the OSF were discussed. However, the dynamic modeling and analysis of the switch wheeled OMRs have not yet been studied, which are the focus of this paper.

In this paper, the dynamic modeling and analysis of a three wheeled omnidirectional mobile robot with MY wheels-II are presented. The detailed dynamic model is derived for the robot, including the motor dynamics. Then the robot dynamic properties are analyzed on the basis of the detailed dynamic model, which show that the robot is a switched nonlinear system. The average dynamic model is derived by simply using the average contact radius in the detailed dynamic model, resulting in a smooth nonlinear dynamic model. The proposed average dynamic model may be used for the modelbased controller design. Finally, the robot dynamic properties and the proposed average dynamic model in predicting the behavior of the detailed dynamic model are shown through open-loop simulations.

The remainder of this paper is organized as follows. In Section II, the detailed dynamic model for a three-wheeled prototype platform is derived. The analysis of the robot dynamics is also presented. The average dynamic model is proposed in Section III. In Section IV, open-loop simulations are presented. Finally, conclusions are drawn in Section V.

978-1-4673-6357-0/13/$31.00 ?2013 IEEE

4860

 =270?

=180? 1?

=90?

45? 22.5?

=0?

(a)

(b)

Y W &2

.

Y

M

&1

Din

O M

6 Dout

X M

Fig. 2. (a) End view of MY wheel-II. (b) Prototype platform.

II. DETAILED DYNAMIC MODELING AND ANALYSIS

A. Detailed Dynamic Modeling

The MY wheel-II mechanism and its end view are shown in Fig. 1(c) and Fig. 2(a), respectively. The wheel consists of two balls of equal diameter on a common shaft and both balls are sliced into four spherical crowns. During the rotation of the main shaft, the two sets of crowns alternatively contact with the ground to realize continuous motion. The two contact points with ground switch between the two sets of crowns whenever the shaft turns 45, and therefore eight switches occur during each turn (see Fig. 2(a)).

The prototype platform is shown in Fig. 2(b), with three MY wheel-II assemblies arranged at a 120 interval angle underneath the steel disk. For a detailed description of MY wheel-II mechanism and the prototype robot, we refer readers to [6].

The dynamic model is derived based on the following assumptions, which are often made in the literature [9], [10]. It is assumed that no slippage is between the wheel and the motion surface. The wheel contact friction forces in the direction perpendicular to the traction force are ignored. The friction forces on the wheel shaft and gear are viscous friction. For the dynamic modeling with static friction model (coulomb and viscous friction), we refer readers to [16] and the references therein. In addition, the motor electric circuit dynamics is neglected.

Two coordinate frames are used in the modeling (Fig. 3): the world coordinate frame {W} which is fixed on the ground and the moving coordinate frame {M} which is fixed on the center of gravity of the robot. The nomenclatures are defined as follows:

World coordinate frame

T

q= x y

Robot location and orientation angle

Moving coordinate frame

VM =

Vx

Vy

T

Robot translational velocity and rota-

tional angular rate expressed in the moving coordinate frame

T

F = Fx Fy Traction force applied to the center of gravity of the robot expressed in the moving coordinate frame

Ti

The traction force of each assembly, i = 1, 2, 3.

&3

O W

X

W

Fig. 3. Coordinate frames of the omnidirectional mobile robot.

T 2b T 2a

Y M F

T 1b

T 1a

O

6

X M

M

T 3a

T 3b

Fig. 4. Force analysis.

Mechanical constants

m

Robot mass

Iv

Robot moment of inertia

Iw

Wheel moment of inertia

R

Wheel radius

Din Inner contact radius

Dout Outer contact radius

n

Gear reduction ratio

The coordinate transformation matrix from the moving coordinate frame to the world coordinate frame is as follows:

W M

R

=

cos sin

0

- sin cos

0

0 0 1

,

(1)

and we get

q

=

W M

RVM

.

(2)

The dynamic properties of the mobile robot can be described with respect to the moving coordinate frame as [9] [10]:

m(V x - Vy) = Fx

m(V y + Vx) = Fy

(3)

Iv? = MI

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where MI is the moment of force around the axis of the robot gravity center. Fx, Fy and MI can be obtained from Fig. 4 :

Fx = -21 T1 - 21T2 + T3

Fy =

3 2

T

1

-

3 2

T2

(4)

MI = T1L1 + T2L2 + T3L3

where Li is the contact radius of each assembly, i = 1, 2, 3.

Li =

Din, Dout ,

if if

-8 +8

n

2

+

<

n 2

i < i

3

8

8++n2n2

n = 0, ?1, ?2, ...

and Ti is the traction force of each assembly, i = 1, 2, 3.

Ti =

Tia, Tib,

i i

f f

8

-

+8 +n2n2 ................
................

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